Problem: Simplify. $i ^ {8}$
Solution: The most important property of the imaginary unit $i$ is that ${i ^ 2} = {-1}$ When this property is plugged into $i ^ 4$ , we get: $i ^ 4 = ({i ^ 2}) ^ 2 = ({-1}) ^ 2 = 1$ So, we can reduce the exponent by multiples of 4 and have the same result. The remainder after dividing 8 by 4 is 0, so $i ^ {8} = i ^ {0}$ Any number but zero to the zeroth power is one. $i ^ 0 = 1$ $i ^ {8} = i ^ {0} = 1$.